An inverse method for the identification of strain-rate sensitivity parameters of sheet steels

Transcription

1 An inverse method for the identification of strain-rate sensitivity parameters of sheet steels M. Avalle, G. Belingardi & M. Gamarino Department of Mechanics, Technical University of Turin, Italy Abstract It is widely accepted that most materials behave differently when submitted to different levels of strain-rate. The strain-rate sensitivity can be modelled by means of constitutive models. The various models proposed in the literature are quite different and criteria for their selection are not available. Furthermore material data are not commonly available because their identification is rather difficult. The aim of this work is to propose a numerical procedure to identify the material parameters needed to take into account for the strain-rate sensitivity. The identification procedure is based on the simulation of an experimental test. Two different constitutive models have been considered. Advantages and limitations of these models are discussed and parameters for the two considered materials were identified. Keywords: strain-rate sensitivity, material properties identification. Introduction Numerical simulation by means of the finite element methods offers nowadays a convenient and feasible tool to design structural components even in very complicated environments such as crash loading. Numerical software codes such as Dyna3D (and its progeny) help in the reduction of time to market and design cost of vehicles, by decreasing the need for real model crash tests that could be substituted by virtual tests. One of the main issues in dynamic high-deformation analysis is the appropriate modelling of the strain-rate effect. The dynamic behaviour of most materials, including metals, differs from the static behaviour [-4] both in terms of strength and strain limits. Strain-rate sensitivity is usually introduced in the mathematical description by means of simple models that in most cases describe 4 WIT Press, ISBN X

2 4 Structures Under Shock and Impact VIII the change from static to dynamic behaviour only in terms of strain-rate dependency. For most materials, including the materials considered in this work, there is a positive trend of the material strength with the strain-rate, as well as a decrease of ductility. Aim of this work is to implement the constitutive models adequate to describe the behaviour of automotive steel into an explicit finite element code, and to identify the model parameters that fit most accurately the experimental results. The experimental test and its model To properly study the behaviour of vehicle structural components subjected to crash, it is important to identify correctly the material properties. This identification must, possibly, be based on experimental test that is significant of the type of loading that the structure will suffer. Therefore, taking into account the typical folding mechanism of collapse, the identification of the material properties and the validation of the numerical models have been performed through a test in which the material is loaded in flexural mode. The experimental tests were performed on square plates mm mm [-7], submitted to bending loads according to the ASTM D4/96 standard (Impact resistance of rigid plastic sheeting or parts by means of a tup). The samples were clamped in a fixing support as shown in figure. The plates of this supporting device allow a free region of 73. mm diameter. Therefore, the sample can be considered as a circular plate clamped along a circular edge. The tup has a mm hemispherical head moved by the actuator of the hydraulic machine in the quasi-static and low-speed tests, and by the impactor of a fallingdart apparatus for the high-speed impact tests. To avoid friction between the head and the specimen, that is difficult to model and adds a source of uncertainty, lubricant was added between the surfaces. Figure : Sample fixing support and numerical model of the sample and tup. 4 WIT Press, ISBN X

3 Structures Under Shock and Impact VIII In figure the finite element mesh used in the simulations is also shown. Taking advantage of double symmetry of the structure to decrease computing time, only a quarter of the plate was modelled. Both the specimen and the tup were modelled by means of shell elements. The material of the specimen was modelled by means of an elastic-plastic material model with isotropic hardening. The impactor tup was modelled with a rigid material. The plate was then constrained along the circular edge with rigid constraints. It was chosen to not model fracture of the material, which seemed computationally not efficient and difficult to manage. 3 Basic mechanical properties of the considered steels The experimental tests were performed on two different materials, a mild steel DC6 EN 3 (formerly FeP6) and a high-strength steel S3 EN 7- (formerly FeE3). The DC6 is hot rolled low carbon steel usually used for deep-drawing in the automotive industry for the car body construction. The S3 steel is a high-strength steel, hot rolled, usually used in highstrength welded structures. In the recent years there has been a growing interest in the use of materials of this type for automotive body constructions. In fact, this type of material can help in reducing structural weight by decreasing the sheet thickness as much as possible. Main material properties of the two steels are reported in table. These data were obtained through some standard tensile tests performed at the beginning of the experimental work in order to obtain the basic stress-strain characteristics to be supplied to the finite element code material for the simulations. Table : Main material properties of the two steels. Material S y (MPa) nominal S r (MPa) nominal E (GPa) measured E p (MPa) measured σ y (MPa) measured DC S3 > Low-speed tests and simulations As a first step, the model was validated through simulation of quasi-static tests. Quasi-static experimental tests were performed by means of a universal hydraulic material testing machine, at a constant loading speed of. mm/s. The described clamping device was used. The load-stroke curves resulting from the quasi-static numerical simulations are compared to the experimental results in figure. There is a very good agreement between the two results, for both materials, at least up to the point at which fracture occurs in the sheet. To quantitatively evaluate the quality of simulation results a performance index was introduced. This was done since a simple qualitative evaluation was 4 WIT Press, ISBN X

4 6 Structures Under Shock and Impact VIII not sufficient to drive an identification algorithm. The used performance index was the global mean square error e calculated as follows: N ( Fi,exp Fi, num ) i e = () N Where F i,exp and F i,num are, respectively, the load values at a same time in the numerical simulation and in an experimental test, and N is the number of experimental sample points. The ratio of the total square error by the number of samples makes the index independent from the number of samples. The evaluation of the global square error performance index was extended from the beginning of the phenomenon up the point for which there is maximum load Sim. num. S3 Sper_S3_s Sper_S3_s Sper_S3_s3 Sper_S3_s4 Sper_S3_s Sper_DC6_s Sper_DC6_s Num_DC6_QS 3 3 Figure : 3 Numerical and experimental load-stroke curves for S3 and DC6 steels Figure 3: Typical load-stroke curve of an impact test. High-speed impact tests and simulations The same mathematical model used to simulate the structure behaviour in the quasi-static case was used for impact simulations. The experimental impact tests were performed under a drop-dart machine, ATS-FAAR Fall-O-Scope. Falling 4 WIT Press, ISBN X

5 Structures Under Shock and Impact VIII 7 height is up to m and the falling mass is kg. Maximum speed is then about 6. m/s, while the maximum potential energy is about 4 J. To better understand the behaviour of the component, the curve shown in figure 3 can be used to detail the load-stroke response. Three different phases can be recognised. In the first phase there is almost linear load-stroke behaviour. The load increases proportionally to the impactor stroke. Since impact excites axial modal vibration of the tup, there are some oscillations of the force value at the very beginning of the curve, these oscillations are meaningless and can be easily filtered out. In the second phase the material is locally subjected to high strains; the material begins to be damaged, especially in the region under the tup. The third phase is characterised by load decrease due to necking and then to fracture. In some experimental tests, this third phase was not reached since with the lower amount of the falling weight kinetic energy, there was not penetration but rebound. Because of uncertainties in the identification of the transition point from the first to the second phase, especially for the cases without complete penetration, it was decided to compute the performance index up to three quarters of the maximum force and not up to the maximum value. In figure 4, the results of two experimental tests at 6.6 m/s are shown. Comparison of the results from quasi-static and dynamic impact tests (in figures and 4) shows significant differences in the behaviour of both materials with the testing velocity. The slope of the load-stroke curves in the first loading and the maximum force values increase passing from the quasi-static to the dynamic cases, whereas the stroke at which fracture occurs decreases. 3 DC6 3 Sper Din Sper Din Sper Din S Sper Din F98 Sper Din F96 Sper Din F94 Figure 4: Experimental test results at 6.6 m/s. 4 WIT Press, ISBN X

6 8 Structures Under Shock and Impact VIII In the following the strain-rate sensitivity will be analysed and modelled by means of the well known strain-rate models proposed by Jones [], Symonds [], and Johnson and Cook [4]. The material parameters for these models will be identified by means of factorial plans aimed to the minimisation of the global square error for the analysed response.. Cowper-Symonds strain-rate model implementation The Cowper-Symonds [,] model has been used to consider strain-rate effects. In the original formulation the strain-rate value only modifies the yield stress and does not influence the slope of the stress-strain curve after yield: σ σ ' ε = + D / q () Table : Cowper-Symonds parameters identification. (a) DC6 D q Legend e < > (b) S3 D q Legend < > In formula () σ is the quasi-static yield-stress of the material, and σ is the yield stress at the strain-rate value ε. The q and D parameters are strain rate e 4 WIT Press, ISBN X

7 Structures Under Shock and Impact VIII 9 material properties; they change from material to material. Aim of the identification procedure was to identify the q and D value for the considered materials. The global square error of the numerical curves with respect to the experimental one is a function of both parameters. The parameters are therefore identified by finding the values that give the minimum global square error. The DC6 deep-drawing steel is generally considered a strain-rate sensitive material, therefore the factorial plan was centred at low D values and high q values. On the contrary the high-strength S3 steel is considered to be a less strain-rate sensitive material, therefore with high values of the D parameter and low values of the q exponent. This is also found in the technical literature [] The analysis was based on the comparison of the experimental and numerical results of the tests in quasi-static and dynamic impact tests at 6.6 m/s. The results of the analysis, in terms of the global square error, are reported in table. Each numerical simulation corresponds to a couple of D and q values. To help understanding the influence of the two parameters, the results are also shown with different grey level of the background cell. White background puts in evidence the best fit results. For DC6 steel the values q = 4. and D = /s have been identified. In table (a) it is visible the greater influence of the D parameter with respect the exponent q Dinamica Dinamica Dinamica 3 Numerica (a) Dinamica Dinamica Dinamica 3 Numerica (b) Figure : Comparison of the load-stroke curves: (a) DC6; (b) S3. For S3 steel the values q =. and D = /s as well as the values D = 8 /s and q = 3 can be identified. Both results put in evidence that the S3 steel is a less strain-sensitive material. There is a quite large plateau in the 4 WIT Press, ISBN X

8 Structures Under Shock and Impact VIII surface describing the influence of both parameters on the global error and two local minima, almost equivalent, can be individuated. The results in terms of load-stroke curve (figure ) are quite satisfactory. The figure shows the comparison of the experimental and numerical curves, for the two materials, and with the identified values of the strain-rate parameters of the Cowper-Symonds law. Global square error, Global q square error, 3, 4 6 q 7 3 D (/s) 3 q D (/s) Figure 6: Surfaces of the global square error: (a) DC6; (b) S3. Finally in figure 6 the results of the factorial analysis are reported as response surfaces showing the influence of the parameters in the graphical form.. Johnson-Cook strain-rate model implementation In 98 Johnson and Cook [4] proposed the following strain-rate model: σ ' = (3) ( ) m n ε + + T 3 A Bε p C ln ε Tmelt 3 In their purposes they would include the effects of work-hardening, strain-rate and temperature. In (3) A is the elastic limit, B and n are the work hardening parameter and exponent, C is the strain-rate parameter, T melt is the temperature parameter (corresponding to the melting point of the material) and m is the temperature exponent. There is still a strain-rate reference constant ε that is usually suggested to choose equal to /s. In our analysis the temperature influence has not been taken into account since it is not important for the current application. For what concerns the basic properties A, B e n that define the quasi-static stress-strain characteristic, they have been obtained from initially performed standard tensile tests. The obtained values are reported in table 4. 4 WIT Press, ISBN X

9 Table 3: Structures Under Shock and Impact VIII Basic material properties. Material A (MPa) B (MPa) n DC S When implementing Johnson-Cook model a slight modification has been introduced. This was done to avoid that the dynamic yield stress could be less than the quasi-static yield stress or even negative, as it is possible by calculation with expression (3) for strain-rate values largely less than /s. The modified law is described as follows: if ε : if ε > : σ ' = n ( A + Bε ) p ' σ = σ = ε + C ln ε T T melt 3 3 m (4) As in the other cases, the identification of the C parameter was obtained by minimizing the global square error. The global square error is shown in figure 7 as a function of the parameter C. The value of the C parameter has been identified as C =.33 for the DC6 steel and C =.74 for the S3 steel. Global square error Global square error 3 4,, 4 3, 3,,,,,,,,4,6,8,3,3,34,36,38,4,,4,6,8,, C C Figure 7: Global square error vs. Johnson-Cook strain-parameter C: (a) DC6; steel (b) S3 steel. 6 Conclusions In the present work two well known strain-rate models have been compared when applied to automotive body construction steels. The assessment of the modelling capability of the two considered models was based on the simulation of an experimental test. The experimental test was a standard lateral punching test obtained by hitting the lateral surface of a circular specimen by a hemispherical tup. 4 WIT Press, ISBN X

10 Structures Under Shock and Impact VIII By means of the same simulation, the suitable values of parameters characteristic of the material could be identified by minimisation of the global square error between the experimental and the numerical load-stroke curves. The Cowper-Symonds model is able to fit adequately both materials in a wide range of loading speeds. The typical shape of the resulting response surface of the global square error over the values of the two searched parameters has been shown and discussed. The Johnson-Cook model too gives results in good accordance with the experimental results. The Johnson-Cook model is attractive due to its simplicity: there is only one structural parameter to be identified. Consequently, the number of necessary simulations is very few Finally, it has been found that the deep-drawing DC6 steel is much more strain-rate sensitive than the high-strength S3 steel. References [] Jones, N., Structural Impact, Cambridge University Press, 989. [] Symonds, P.S., Survey of methods of analysis for plastic deformation of structures under dynamic loading, Report No. BU/NSRDC, Brown University, 967. [3] Jones, N., Some comments on the modelling of material properties for dynamic structural plasticity, Proc. of the Int. Conf. Mechanical on Properties of Materials at High Rates of Strain, Oxford, pp.43-44, 989. [4] Johnson, G.R & Cook, W.H., Fracture Characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Engineering Fracture Mechanics, (), pp. 3-48, 98. [] Avalle, M., Belingardi, G., Vadori, R. & Masciocco, G., Caratterizzazione dinamica del comportamento flessionale di acciai da stampaggio, Proc. of the XXVIII AIAS National Conference, Vicenza, pp , 999. [6] Belingardi, G., Fornara, A.A. & Masciocco, G., Valutazione della sensibilità allo strain-rate di acciai altoresistenziali, Proc. of the XXIX AIAS National Conference, Lucca, pp. 93-,. [7] Avalle, M., Belingardi, G., Vadori, R. & Masciocco, G., Characterization of the strain rate sensitivity in the dynamic bending behavior of mild steel plates, Proc. of the EUROMAT, Tours, pp. -,. [8] Gamarino, M., Avalle, M. & Vadori, R., Modelli della sensibilità allo strain-rate e identificazione dei parametri caratteristici di lamiere di acciaio, Proc. of the XXXII AIAS National Conference, Salerno, 3. 4 WIT Press, ISBN X